澳门大学10-11,09-10入学考试试题 及 参考答案.pdf

Past Examination Papers 2009 2010 2010 2011 62 62 2010 2011 學年 入學考試試題 第一部分 七題全部作答 1 解 等式 15xx 8分 2 設 x x e e xf 1 2 a 證明 xf 是增函 3分 b 求 xf 的反函 1 xf 的表達式 4分 3 a 證明 2 3 tan31 tantan3 3tan 提示 BA BA BA tantan1 tantan tan 3分 b 求最小的正 使得 tan 1 3tan 答案以 arctan 表示 4分 4 a 求下 方程組的通解 542 1 zyx zyx 3分 b 用 a 的通解 求下 方程組的實解 3 542 1 zy zyx zyx 4分 Syllabus 2011 2012 6363 B C O P 圖I 5 設 1 nn a 為一正 使得 n aa aa nn n 321 6 43 2 1 用 學歸納法 證明 nnan321 13 8分 6 a 用恆等式 sin sin sincos2BABABA 證明 sin 5sin sin 4cos 2cos cos2 推導出 0 3 4 cos 3 2 cos cos 4分 b 如圖I ABC是等邊三角形 圓ABC的圓心為O 其 半 徑 為1 P為 圓 上 一 點 證 明 6 222 PCPBPA 提示 設 AOP 並用 a 的結果 4分 7 一袋中有球九個 分別標有1 至 9 號 隨機同時抽出三個球 問 a 三個抽出的球的標號的乘積是偶 的概 是多少 3分 b 三個抽出的球的標號的和是偶 的概 是多少 4分 第二部分 任擇三題作答 每題十 分 8 a 設 m 為正整 在 m mxx 21 2 的展開式中 2 x的係 為65 求 m 的值 7 分 b 設 2 n 為正整 及對於 nr 0 r n C 是在 n x 1 的展開式中 r x 的係 Past Examination Papers 2009 2010 2010 2011 64 64 i 證明 r n r n CrnnC 1 2 分 ii 證明 2 22221 1 12 2 n n CnCC n nnn 提示 考慮 nnn xxnxn 1 1 1 112 7 分 9 如圖II ABC是等邊三角形 其邊長為1 BCDE是梯形 BECD 2 BCD 2 CD 平面ABC垂直於平面BCDE a 求 AD 與平面BCDE所成的角 5 分 b 設 2 AED i 以 BE 表 AE 及 ED 由此 證明 2 2 BE 7 分 ii 求點C與平面ADE的距 提示 考慮三棱錐ACDE的體積 4 分 A B C D E 圖II Syllabus 2011 2012 6565 10 已知橢圓 1 2 2 2 y x E a 直線 cmxyL 與橢圓 E 相 證明 12 22 mc 推導出 直線 L 過點 kh 則 0 1 2 2 222 khkmmh 5分 b 求過點 2 2 並與橢圓 E 相 的 直線的方程 3分 c 設 2 hP 為直線 2 y 上一點 其中2 h 設過點 P 並與橢 圓 E 相 的 直線 1 L 及 2 L 的斜 分別為 1 m 及 2 m i 求 21 mm 及 21m m 2分 ii 1 L 及 2 L 的夾角是 4 求點P 6分 11 a 設 xxxy24152 23 i 求 dx dy 及 2 2 dx yd 2分 ii 繪出曲線 xxxy24152 23 並在圖中把局部極大點 局部極 小點和拐點標出 6分 iii 求所有可能的 k 值使得方程 024152 23 kxxx 有重根 提示 考慮 ii 的圖像 2分 b 求由曲線 xxxy24152 23 與 曲線 xxy2411 2 所包圍的區 域的面積 6分 12 a 用棣美弗定 證明 cos5cos20cos165cos 35 6 分 b 用 a 的結果 求所有可能的 值 其中 0 使得 cos 是 方程 052016 24 xx 的根 4 分 Past Examination Papers 2009 2010 2010 2011 66 66 c 用 b 的結果 證明 16 1 10 9 cos1 10 7 cos1 10 3 cos1 10 cos1 由此 或用其他方法 求 10 3 sin 10 sin 6 分 全卷完 Syllabus 2011 2012 6767 2010 2011 ADMISSION EXAMINATION PAPER Section I Answer all 7 questions 1 Solve the inequality 15xx 8 marks 2 Let x x e e xf 1 2 a Show that xf is an increasing function 3 marks b Find the inverse function 1 xf of xf in closed formula 4 marks 3 a Show that 2 3 tan31 tantan3 3tan Hint BA BA BA tantan1 tantan tan 3 marks b Find the smallest positive such that tan 1 3tan Express your answer in terms of arctan 4 marks 4 a Find the general solution of the following system of equations 542 1 zyx zyx 3 marks b Using the general solution found in a find the real solutions of the system 3 542 1 zy zyx zyx 4 marks Past Examination Papers 2009 2010 2010 2011 68 68 A B C O P Fig I 5 Let 1 nn a be a sequence of positive numbers such that 6 43 2 1 nn n aa aa 3 2 1 n Using mathematical induction prove that nnan321 13 8 marks 6 a Using the identity sin sin sincos2BABABA show that sin 5sin sin 4cos 2cos cos2 Deduce that 0 3 4 cos 3 2 cos cos 4 marks b In Figure I ABC is an equilateral triangle Circle ABC has O as its center and radius 1 P is a point on the circle Show that 6 222 PCPBPA Hint Let AOP and use the result in a 4 marks 7 There are nine balls in a bag The balls are labeled from 1 to 9 If three balls are drawn randomly at the same time what is the probability that a the product of the numbers of the three balls drawn is even 3 marks b the sum of the numbers of the three balls drawn is even 4 marks Syllabus 2011 2012 6969 Section II Answer any three questions Each carries 16 marks 8 a Let m be a positive integer Suppose that in the expansion of m mxx 21 2 the coefficient of 2 x is 65 Find the value of m 7 marks b Let 2 n be a positive integer and for nr 0 r n C denote the coefficient of r x in the expansion of n x 1 i Show that r n r n CrnnC 1 2 marks ii Show that 2 22221 1 12 2 n n CnCC n nnn Hint Consider nnn xxnxn 1 1 1 112 7 marks 9 In Figure II ABC is an equilateral triangle with side length 1 BCDE is a trapezium BECD 2 BCD and 2 CD The planes ABC and BCDE are perpendicular a Find the angle between AD and the plane BCDE 5 marks A B C D E Fig II Past Examination Papers 2009 2010 2010 2011 70 70 b Suppose 2 AED i Express AE and ED in terms of BE Hence show that 2 2 BE 7 marks ii Find the distance between the point C and the plane ADE Hint Consider the volume of the triangular pyramid ACDE 4 marks 10 Given the ellipse 1 2 2 2 y x E a If the line cmxyL is tangent to ellipse E show that 12 22 mc Deduce that if the line L passes through the point kh then 0 1 2 2 222 khkmmh 5 marks b Find the equations of the two lines that pass through the point 2 2 and are tangent to ellipse E 3 marks c Let 2 hP be a point on the line 2 y with 2 h Suppose the slopes of the two lines 1 L and 2 L that pass through point P and tangent to ellipse E are 1 m and 2 m respectively i Find 21 mm and 21m m 2 marks ii If the angle between the lines 1 L and 2 L is 4 find the pointP 6 marks 11 a Let xxxy24152 23 i Find dx dy and 2 2 dx yd 2 marks ii Sketch the curve xxxy24152 23 Mark in the graph the local Syllabus 2011 2012 7171 maximum point local minimum point and inflection point 6 marks iii Find all the possible values of k such that the equation 024152 23 kxxx has double root Hint Consider the graph in ii 2 marks b Find the area of the region bounded by the curves xxxy24152 23 and xxy2411 2 6 marks 12 a Using De Moivre s theorem show that cos5cos20cos165cos 35 6 marks b Using the result in a find all possible values of with 0 such that cos is a root of the equation 052016 24 xx 4 marks c Using the result in b show that 16 1 10 9 cos1 10 7 cos1 10 3 cos1 10 cos1 Hence or otherwise find 10 3 sin 10 sin 6 marks End of paper Past Examination Papers 2009 2010 2010 2011 72 72 2010 2011 學年 參考答案 MODEL ANSWER 1 15 x or 2 171 x 2 a For vu 0 1 vu vuvu e eee vfuf b 2 4 ln 2 1 xx xf 3 a Write 3tan as 2tan and apply the formula given in the hint b 83arctan 4 a 21 2tztytx t is any real number b 3 2 3 7 3 8 or 3 4 3 5 3 2 5 When 1 n 6 43 1 2 1 1 aa a implies 4 1 a 1 1 a is rejected Suppose the result is true for all positive integer nk Then 6 43 2 13 4 6 43 13 4 6 43 1 2 1 1 1 2 1 1 1 2 1 11 nn n nn n nn nn aa a n n aa an aa aaa Solving 1 1 3 1 nan 13 1 nan is rejected 6 a For 4 2 0 k we have 1 sin 1 sin sin cos 2 kkk The result follows Putting 3 the result follows Syllabus 2011 2012 7373 b Using cosine rule cos22 2 AP 3 2 cos 22 2 BP and 3 4 cos 22 2 CP The result follows from a 7 a 42 37 b 21 11 8 a In the expansion of m mxx 21 2 the coefficient of 2 x is mm23 2 Consequently 5 m b i r n r n Crn rnr n rn rnr n nnC 1 1 1 ii In the expansion of 12 1 n xn the coefficient of n x is 2 1 12 n n In the expansion of nn xxn 1 1 1 by using b i the coefficient of n x is 22221 2 n nnn CnCC The result follows 9 a Let M be the mid point of BC Then BCAM as ABC is an equilateral triangle Since BCDEABC we get MDAM Hence the angle required isADM As 2 3 AM and 2 3 MD we get 63 3 arctan ADM b i Given 2 BCD andCDBE So 2 CBE Together withBCDEABC we get 2 ABE Thus 2 1 BEAE From trapeziumBCDE we get 2 2 1 BEED From 2 BCD andBCDEABC we have 2 ACD Hence 3 AD As 2 AED we have 222 ADEDAE Hence 2 2 BE Past Examination Papers 2009 2010 2010 2011 74 74 ii Volume of CDEA is 12 6 Volume of ADEC is 4 d where d is the distance between point C and plane ADE Hence 3 6 d 10 a At intersection of ellipse E and line L we have 0 22 4 12 222 cmcxxm For tangency the discriminant is 0 Thus 12 22 mc If the line L passes through kh we have cmhk Thus 12 22 mmhk and the result follows b Using a one tangent is 2 8 23 2 xy The other tangent is 2 x c i 2 4 2 21 h h mm and 2 3 2 21 h mm ii From the given condition 1 1 2 21 21 mm mm i e 1 1 4 2 21 21 2 21 mm mmmm Using i we have 0232 24 hh Hence 241 h 11 a i 24306 2 xx dx dy 3012 2 2 x dx yd ii 4or 10 xx dx dy 2 5 0 2 2 x dx yd The curve has 1 11 as a local maximum point 4 16 as a local minimum point and 2 5 2 5 as an inflection point iii 16 k or 11 k c 3 8 24152 2411 2 0 232 dxxxxx 12 a By comparing the real parts of both sides of 5sin5cos sin cos 5 ii the result follows b We need 05cos and 0cos Hence 10 9 10 7 10 3 10 Syllabus 2011 2012 7575 c By b 10 9 cos 10 7 cos 10 3 cos 10 cos 1652016 24 xxxxxx Putting 1 x the result of the first part follows Note that 10 3 cos 10 7 cos and 10 cos 10 9 cos We get from above that 16 1 10 3 cos1 10 cos1 22 Hence 4 1 10 3 sin 10 sin Past Examination Papers 2009 2010 2010 2011 76 76 2009 2010 學年 入學考試試題 第一部分 七題全部作答 1 解方程 042log3log3 1010 xx 6分 2 設 和 為方程 0 22 3 22 kkkxx 的根 其中 k 為常 a 證明 和 為 相同的實 2分 b 以 k 表 2 2 3分 c 設 2 求 k 的取值範圍 3分 3 a 設 1tan 4cottan 求 cos 4分 b 證明恆等式 A BB BAB ABA 2 cos cossin tan tan1 tan tan 4分 4 a 設0 x 證明 012 1 2 x x x 3分 b 用 學歸納法及 a 的結果 證明對任意正整 n 212 1 3 1 2 1 1 n n 4分 5 設下 方程組 E 有非 解 035 03 3 0 pzyx zyx zyx E 其中 p 為常 a 求 p 的值 3分 b 求 E 的通解 5分 6 設 n 為正整 在 n x 1 的展開式中 4 x 5 x及 6 x 的係 按序 為一等差 中的 續三項 求 n 的值 7分 7 一袋中有球10個 分別標有 字1 至 10 隨機同時抽出4個球 問 a 在抽出的球中最少有 個球的 字 小於5的概 是多少 4分 b 在抽出的球中任意 個球的 字之和都 是9的概 是多少 4 分 Syllabus 2011 2012 7777 第二部分 任擇三題作答 每題十 分 8 已知拋物線 2 4 xyP 設 4 2 aaA 為 P 上一點 其中 0 a a 直線 cmxy 與拋物線 P 相 證明 016 2 cm 由此 或 用其他方法 推導出拋物線 P 在點 A 的 線的斜 為 a8 5分 b 設拋物線P在點A的 線及法線分別與y 軸相交於點 H 和點 K 證 明 HK 的中點為點 16 1 0 F 5分 c 設 C 是以點 F 為圓心並通過點 A 的圓 證明拋物線 P 在點 A 的 線與圓 C 在點 A 的 線的夾角為 8 tan 1 a 6分 9 a 如圖I 平面 與 相交於直線l 點 A 和點 B 分別在平面 及平 面 之上 點 P 和點 Q 分別為點A 及點B到直線 l 的垂足 設點X 在平 面 之 上 使 得 QBPX 及 PQXB i 證明 PQBX 為一矩形 2分 ii 證明 2 AXB 3分 iii 設 為平面 與平面 所夾的二面角 證明 2 cos 2222 BQAP ABPQBQAP 3分 b 如圖II ABCD 是三棱錐 1 BC 2 ABCDABDAC 6 ACD 及 4 BCD 用 a iii 的結果 或用其他方法 求二面角 BDCA 8分 Q B P A 圖I X A B C A 圖II Past Examination Papers 2009 2010 2010 2011 78 78 10 a 如圖III 正圓錐的高及底半徑均為 1 圓 錐內接有半徑為 x 的正圓柱 設 xV 為該 圓柱的體積 i 證明 32 xxxV 10 x 2分 ii 求 dx dV 及 2 2 dx Vd 2分 iii 繪出 xV 的圖像 並在圖中把局 部極大點 局部極小點和拐點標出 6分 b 設10 a 求由曲線 32 xxy 及 2 axy 所包圍的面積 6分 11 設 1 i 為純虛的複 使得1 2 i a 設 20 及 證明 2 tan sincos1 sin1cos i i i 4分 b 給出方程 1 1 1 n z z 其中 n 為正整 i 證明當 n 為偶 時 方程的解為 n k i 2 12 tan 1 1 0 nk 5分 ii 當 n 為奇 時 方程的解為何 提示 方程只有 1 n 個解 2分 c 用 b i 的 結 果 或 用 其 他 方 法 解 方 程 0 1 1 1 8428 zzz 提示 44 1212 8428 1 1 1 1 1 1 1 zz zz zzz 5分 12 a 設 n 為正整 n n nxxxS 2 2 其中 1 x 考慮 nn xSS 證明 x nx x xx S nn n 1 1 1 1 2 4分 b 設 1 nn a 為一等比 其公比 1 r 且 21 321 aaa 及 216 321 aaa i 求此等比 的通項 n a 5分 ii 用 a 的結果 求 n a n aa aaa 21 21 2 1 n 7分 全卷完 圖III x Syllabus 2011 2012 7979 2009 2010 ADMISSION EXAMINATION PAPER Section I Answer all 7 questions 1 Solve the equation 042log3log3 1010 xx 6 marks 2 Let and be the roots of the equation 0 22 3 22 kkkxx where k is a constant a Show that and are real and distinct numbers 2 marks b Express 2 2 in terms of k 3 marks c Suppose 2 Find the range of k 3 marks 3 a Let 1tan If 4cottan find cos 4 marks b Show the following identity A BB BAB ABA 2 cos cossin tan tan1 tan tan 4 marks 4 a Let0 x Show that 012 1 2 x x x 3 marks b Prove by mathematical induction and the result in a that for any positive integer n 212 1 3 1 2 1 1 n n 4 marks 5 Given that the system of equations E below has nonzero solutions 035 03 3 0 pzyx zyx zyx E where p is a constant a Find the value of p 3 marks b Find the general solution of E 5 marks 6 Let n be a positive integer Suppose that in the expansion of n x 1 the coefficients of 4 x 5 x and 6 x are in arithmetic progression Find the value s of n 7 marks 7 A bag contains 10 balls labeled with numbers from 1 to 10 If 4 balls are drawn randomly at the same time what is the probability that a at least two balls with numbers not less than 5 are drawn 4 marks b among the balls drawn the sum of the numbers of any two balls is not 9 4 marks Past Examination Papers 2009 2010 2010 2011 80 80 Section II Answer any 3 questions Each carries 16 marks 8 Given parabola 2 4 xyP Let 4 2 aaA be a point on P where 0 a a If straight line cmxy is tangent to P show that 016 2 cm Hence or otherwise deduce that the slope of the tangent line of P at A is a8 5 marks b Suppose that the tangent line and normal line of P at A intersect with the y axis at points H and K respectively Show that the mid point of HK is 16 1 0 F 5 marks c Let C be the circle centered at F and passing through A Show that the angle between the tangent line of P at A and the tangent line of C at A is 8 tan 1 a 6 marks 9 a In figure I two planes and intersect at line l Points A and B are in and respectively Points P and Q are the feet of perpendicular from points A and B to l respectively Let X be a point on such that QBPX and PQXB i Show that PQBX is a rectangle 2marks ii Show that 2 AXB 3marks iii Let be the angle between the two planes and Show that 2 cos 2222 BQAP ABPQBQAP 3 marks b In figure II ABCD is a triangular pyramid 1 BC 2 ABCDABDAC 6 ACD and 4 BCD Using the result in a iii or otherwise find the dihedral angle BDCA 8 marks Q B P A Fig I X D B C A Fig II Syllabus 2011 2012 8181 10 a In figure III the height and base radius of the right circular cone are both 1 A right circular cylinder with radius x is inscribed in the cone Let xV be the volume of the cylinder i Show that 32 xxxV 10 x 2 marks ii Find dx dV and 2 2 dx Vd 2 marks iii Sketch the graph of xV and mark in the graph the local maximum point local minimum point and inflection point 6 marks b Let 10 a Find the area bounded by the curves 32 xxy and 2 axy 6 marks 11 Let 1 i be the pure imaginary complex number such that 1 2 i a Suppose 20 and Show that 2 tan sincos1 sin1cos i i i 4 marks b Given equation 1 1 1 n z z where n is a positive integer i Prove that when n is even the solutions of the equation are n k i 2 12 tan 1 1 0 nk 5 marks ii What are the solutions of the equation when n is odd Hint The equation has only1 n solutions 2 marks c Using the result in b i or otherwise solve the equation 0 1 1 1 8428 zzz Hint 44 1212 8428 1 1 1 1 1 1 1 zz zz zzz 5 marks 12 a Let n be a positive integer and n n nxxxS 2 2 where 1 x By considering nn xSS show that x nx x xx S nn n 1 1 1 1 2 4 marks b Let 1 nn a be a geometric progression with common ratio 1 r and suppose 21 321 aaa and 216 321 aaa i Find the general term n a of the geometric progression 5 marks ii Using the result in a find n a n aa aaa 21 21 2 1 n 7 marks End of paper Fig III x Past Examination Papers 2009 2010 2010 2011 82 82 2009 2010 學年 參考答案 MODEL ANSWER Outline of solutions Math A 2009 10 1 100 x 2 a The discriminant equals 4 2 2 k which is positive b 272 2 kk c 4 337 4 337 k 3 a 2 32 b By direct calculation using BA BA BA tantan1 tantan tan 4 a 0 1 12 1 2 2 x xx x x x b The key steps are 222 1 1 212 1 1 1 1 k k k kk 5 a 7 p b tz t y t x 2 3 2 where t is any real number 6 7 or 14 7 a 42 37 b 105 52 8 a